378 
Now multiplying both sides of (5) by Q, or w+ ix + jy + hz, 
we have 
Q™ = wWr- Vn (a +y? +2) + (Wn t+ W,) (ix + jy + hz). 
Comparing this with (6), we have 
Waa = ww, —rVq 
Va = wVn ae W,, 
a pair of simultaneous equations of finite differences. From 
these we readily deduce 
Was — 20 Wry + (w? + 7?) W, = 0, 
Vaz — 2wVag + (w? +77) V, = 09. 
The complete integrals of which are 
W,=c(w+ry -1)"+e(w-ry -1)’, (7) 
Vi,=bwiry -1)"+0(w-ry - 1). (8) 
c, ¢, b, b', being arbitrary constants. 
‘“‘To determine the values of these constants let n = 0; 
whence from (5) it is evident that W,=w, V,=0; values 
which, substituted in (7) and (8), give 
l=cied, 
0=5b+8. 
“ Again, let x =1, whence substituting in (5) for Q its 
value w + ia + jy + kz, we find W, = w, V, = 1, and employ- 
ing these values in (7) and (8), we have 
w=c(wtry -1)+ce(w-ry -1), 
l=b(@wiry -1)+0(w-ry -1). 
From these four equations, we find 
Ae let gee Us in) cols 
Dai) Ts BF Mardiodeal Gil hind (Gay Sal 
} where 7? = 2? + y? + 2, 
c= 3 
2 
Substituting these values in (7) and (8), and the resulting 
values of W,, and V, in (5), we have 
ee ee ee 
Te ae 
5 Codey Tea ty AU" (ie + iy + he). (9) 
